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The Quarterly Journal of Mathematics
, Volume Advance Article – Apr 24, 2018

31 pages

/lp/ou_press/on-stably-trivial-spin-torsors-over-low-dimensional-schemes-BJEqjaTsXF

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com
- ISSN
- 0033-5606
- eISSN
- 1464-3847
- D.O.I.
- 10.1093/qmath/hay022
- Publisher site
- See Article on Publisher Site

Abstract The paper discusses stably trivial torsors for spin and orthogonal groups over smooth affine schemes over infinite perfect fields of characteristic unequal to 2. We give a complete description of all the invariants relevant for the classification of such objects over schemes of dimension at most 3, along with many examples. The results are based on the A1-representability theorem for torsors and transfer of known computations of A1-homotopy sheaves along the sporadic isomorphisms to spin groups. 1. Introduction The main point of the paper is to study the classification of stably trivial orthogonal (and spin) bundles over low-dimensional schemes. This is essentially the question how large the difference is between the Grothendieck–Witt ring and the actual isometry classification of quadratic forms over rings. Over fields, Witt’s cancellation theorem tells us that the monoid of isometry classes is cancellative and, therefore, embeds into its group completion. Over commutative rings of higher dimension, this is no longer true, and the present investigation concerns exactly this failure of cancellation for quadratic forms. While most of the contemporary work on quadratic forms is related to stable theories (hermitian K-theory or higher Grothendieck–Witt groups), unstable questions related to the actual isometry classification seem to have mostly been neglected (except of course over rings of integers in local or global fields). The point of the paper is to show that homotopical methods can also be applied to study the isometry classification of quadratic forms over schemes: if we restrict our attention to smooth affine schemes over infinite perfect fields of characteristic ≠2, then the A1-representability result of [6] allows to translate questions concerning classification of rationally hyperbolic quadratic forms into questions of obstruction-theoretic classification of morphisms into classifying spaces of orthogonal groups. Knowledge of A1-homotopy groups of the relevant classifying spaces can then be translated to classification results for rationally hyperbolic quadratic forms (and in particular stably trivial orthogonal bundles). Over smooth affine schemes of dimension ≤3, where only the knowledge of A1-homotopy sheaves up to π3A1 is required, we can actually give a complete description of the relevant invariants entering the classification of rationally hyperbolic forms. This is done via the classical sporadic isomorphisms for low-dimensional orthogonal resp. spin groups and the known A1-homotopy computations for the related groups. Surprisingly, the identification of the orthogonal stabilization maps under the sporadic isomorphisms does not seem to be easy to find in the literature, necessitating a slightly extended discussion of the sporadic isomorphisms. The results can be used to produce many examples of stably trivial spin torsors over various varieties, cf. Section 5. We also discuss the relation between classification of stably trivial spin torsors and quadratic bundles, which leads to a number of explicit examples of stably trivial quadratic bundles, cf. Section 6. The most concise formulation of the combined results of the paper is the following: Theorem 1.1. Let kbe an infinite perfect field of characteristic ≠2, let X=SpecAbe a smooth affine scheme of dimension ≤3over kand let (P,ϕ)be a rationally hyperbolic quadratic form over Aof rank nadmitting a lift of structure group to Spin(n). The only stable invariant of (P,ϕ)is the second Chern class. Here the second Chern class of a quadratic form can be defined by stabilizing (P,ϕ) by adding hyperbolic planes to get a quadratic form of rank ≥7 and then taking the second Chern class of the underlying projective bundle. While the above result identifies a single stable invariant, there are various other unstable invariants leading to various sources of examples of stably trivial quadratic bundles resp. spin torsors. For the group SO(3), stabilization induces multiplication by 2 on the second Chern class which allows for examples related to 2-torsion in Chow groups; for SO(4), there are actually two Chern classes appearing in CH2 leading to a great number of non-trivial stably trivial torsors. Since some of the low-dimensional spin groups are symplectic, there are examples of stably trivial spin torsors related to orientation information in CH˜2. Finally, there are various invariants arising from non-trivial π3A1 which all become trivial after stabilization, because π3A1BSpin(∞)≅K3ind is invisible in the torsor classification. These invariants give rise to interesting examples of stably trivial quadratic forms in a variety of situations. Since the examples arise via the sporadic isomorphisms, they can all be constructed very explicitly. Finally, it should be pointed out here that stably trivial quadratic forms refer here to Nisnevich-locally trivial torsors with structure group O(n) which become trivial upon stabilization to O(∞). This is different from the usual notion of stably hyperbolic quadratic forms, which are quadratic forms which upon adding a hyperbolic form associated to a projective module become isomorphic to the hyperbolic form associated to a projective module. A theorem proved by Ojanguren and Pardon, cf. [11, Section VIII.2], states that over an integral affine scheme of dimension ≤3, any rationally hyperbolic quadratic form is stably hyperbolic. In particular, all of the examples discussed in this paper eventually (after stabilization) arise via the hyperbolic functor from vector bundles. What we provide here is the isometry classification of such things when the underlying scheme is smooth affine over an infinite field of characteristic unequal to 2. The isometry classification actually allows to give examples of non-hyperbolic (stably hyperbolic by the above) forms of rank 4, cf. Example 6.9. 1.1. Conventions In this paper, k always is an infinite perfect field of characteristic unequal to 2. We consider smooth affine schemes X=SpecA over k and are interested in classification results for quadratic forms over the ring A. 2. Preliminaries on quadratic forms This section provides a short recollection on the relevant facts concerning quadratic forms. Most of what is recalled below are standard definitions which can be found in any textbook, cf. for example [11]. Definition 2.1 Let k be a field of characteristic unequal to 2. A quadratic form over a commutative k-algebra A is given by a finitely generated projective A-module P together with a map ϕ:P→A such that for each a∈A and x∈P, we have ϕ(ax)=a2ϕ(x) and Bϕ(x,y)=ϕ(x+y)−ϕ(x)−ϕ(y) is a symmetric bilinear form Bϕ∈Sym2(P∨). The rank of the quadratic form is defined to be the rank of the underlying projective module P. A quadratic form (P,ϕ) is non-singular or non-degenerate if the morphism P→P∨:x↦Bϕ(x,−) is an isomorphism. An element x∈P is called isotropic if ϕ(x)=0. A morphism f:(P1,ϕ1)→(P2,ϕ2) of quadratic forms is an A-linear map f:P1→P2 such that ϕ2(f(x))=ϕ1(x) for all x∈P1. An isomorphism of quadratic forms is also called isometry. The automorphism group of a quadratic form is called the orthogonal group of the quadratic form. Given two quadratic forms (P1,ϕ1) and (P2,ϕ2), there is a quadratic form (P1,ϕ1)⊥(P2,ϕ2)≔(P1⊕P2,ϕ1+ϕ2)called the orthogonal sum. Example 2.2 Let A be a commutative ring and P be a finitely generated projective module. Then there is a quadratic form whose underlying module is P⊕P∨, equipped with the evaluation form ev:(x,f)↦f(x). The quadratic form (P⊕P∨,ev) is called the hyperbolic space associated to the projective module P. In the special case where P=A is the free module of rank 1, this is called the hyperbolic plane H over A. The following are the standard hyperbolicity notions from quadratic form theory, cf. [11, Section VIII.2]. Definition 2.3 A quadratic form is called hyperbolic, if it is isometric to H(P) for some projective module P. A quadratic form (P,ϕ) is called stably hyperbolic if there exists a projective module Q such that (P,ϕ)⊥H(Q) is hyperbolic. A quadratic form (P,ϕ) over an integral domain A is called rationally hyperbolic if (P,ϕ)⊗AFrac(A) is hyperbolic. Remark 2.4 Note that stably hyperbolic forms are then necessarily of even rank, and stably hyperbolic forms are those that become 0 in the Witt ring. For the purposes of the present paper, we will also be interested in stricter notions of stable triviality of quadratic forms: Definition 2.5 A quadratic form (P,ϕ) is called stably trivial if it becomes isometric to one of the split forms H⊥n or H⊥n⊥(A,a↦a2) after adding sufficiently many hyperbolic planes. The stably trivial forms are those which represent classes in Z·[H]⊕Z·[(A,a↦a2)]⊆GW(A) in the Grothendieck–Witt ring of A. The classical Witt cancellation theorem implies in particular that the notions of hyperbolic, stably trivial and stably hyperbolic all agree over fields: Proposition 2.6 (Witt cancellation theorem). Let kbe a field of characteristic ≠2and let (V1,ϕ1)and (V2,ϕ2)be two quadratic forms over k. If (V1,ϕ1)⊕H≅(V2,ϕ2)⊕Hthen (V1,ϕ1)≅(V2,ϕ2). In particular, a stably hyperbolic form is hyperbolic. Remark 2.7 All quadratic forms over fields considered in this paper are hyperbolic or of the form H⊥n⊕(A,a↦a2). In abuse of notation, the group SO(n) will always denote the special orthogonal group associated to the split form of rank n. I apologize to anyone who might be offended by this. Next, we will have a look at torsors under the groups O(n) and SO(n) and how they are related. In the end, we want to represent quadratic forms as suitable equivalence classes of spin torsors because the spin groups are easier to handle with the A1-homotopy methods, but the discussion of the relation between torsors for orthogonal and spin groups is deferred to Section 6. First, we note that étale local triviality of quadratic forms implies that they can be viewed as torsors for the split orthogonal groups. Proposition 2.8 Let kbe a field of characteristic unequal to 2, let X=SpecAbe a k-scheme and denote by O(n)the orthogonal group associated to the hyperbolic quadratic form on kn. There is a functorial bijective correspondence between the set of isometry classes of quadratic forms over Aand the set Hét1(X,O(n))of isomorphism classes of O(n)-torsors over X. Proof A quadratic space is locally trivial in the étale topology by [19, Corollary 1.2]. The remaining argument is standard. Given a quadratic form, we can find an étale cover ⨆iUi→X over which the form trivializes and the transition morphisms are isometries. This implies that we get a morphism Cˇ(U/X)→BétO(n). Conversely, the cocycle condition implies that the locally trivial pieces can be glued to a quadratic form. Then one can check that simplicial homotopies correspond to globally defined isometries.□ There is a group extension 1→SO(n)→O(n)→μ2→0 which implies that the natural map BétSO(n)→BétO(n) is a degree 2 étale covering. Any quadratic form (P,ϕ) over X induces a degree 2 étale covering X˜→X, by pullback of the above along the classifying map X→BétO(n). This is the orientation covering for the quadratic form (P,ϕ) whose class in Hét1(X,μ2) is the first Stiefel–Whitney class w1(P,ϕ). We say that a quadratic form is orientable if its orientation cover is the trivial degree 2 étale map id⊔id:X⊔X→X. A choice of lift of X→BétO(n) to BétSO(n) is called an orientation. Proposition 2.9 Let kbe a field of characteristic unequal to 2, let X=SpecAbe a k-scheme and denote by SO(n)the special orthogonal group associated to the hyperbolic quadratic form on kn. There is a functorial bijective correspondence between the set of isometry classes of orientable quadratic forms over Aand the set Hét1(X,SO(n))of isomorphism classes of SO(n)-torsors over X. Proof By what was said before, the orientability is the necessary and sufficient condition for lifting. The next claim is that there is effectively no choice for the lift. Consider the relevant homotopy sequence in the homotopy theory of simplicial sheaves on the big étale site [X,O(n)]→[X,μ2]→[X,BétSO(n)]→[X,BétO(n)]whose exactness essentially states that the choice of lifts is the choice of orientations, up to isometries. The first map, induced by the projection O(n)→μ2, is a surjection, saying that all orientations are equivalent up to isometry. Hence, we get an injection of orientable forms into all forms. □ The functor X↦Hét1(X,SO(n)) mapping a (smooth) scheme to the pointed set of isometry classes of orientable quadratic forms of rank n is not representable in A1-homotopy, due for example to Parimala’s examples of non-trivial quadratic forms of rank 4 on the affine plane AR2, cf. [16]. However, as we recall later, the functor X↦HNis1(X,SO(n)) is A1-representable, which is why we are interested in rationally hyperbolic forms here. Proposition 2.10 Let kbe an infinite field of characteristic unequal to 2, let X=SpecAbe a smooth affine k-scheme and denote by SO(n)the special orthogonal group associated to the hyperbolic quadratic form on kn. The bijection of Proposition2.9restricts to a bijection from the isometry classes of rationally hyperbolic quadratic forms to the set HNis1(X,SO(n))of rationally trivial SO(n)-torsors. Moreover, this bijection restricts to an injection from stably trivial forms into rationally trivial SO(n)-torsors. Proof We can assume X irreducible. Then the orientation cover of a rationally hyperbolic quadratic form is rationally trivial. But a finite étale map onto a smooth scheme which has a rational section is already trivial. Therefore, a rationally hyperbolic quadratic form over X is orientable. Now the restriction of the bijection from Proposition 2.9 to rationally hyperbolic quadratic forms follows basically from Nisnevich’s theorem identifying HNis1 as torsors with a rational section. So we are left with proving the statement about stably trivial forms. We can again assume that X is irreducible. By extension of scalars, a stably trivial form on A gives rise to a stably hyperbolic form on the field of fractions Frac(A). By Witt cancellation, the form on Frac(A) is hyperbolic. By functoriality of the correspondence in Proposition 2.9, the associated SO(n)-torsor over X is rationally trivial.□ 3. Recollections on A1-homotopy theory In this section, we recall some basics of A1-homotopy theory, in particular the representability result and results of A1-obstruction theory. We assume that the reader is familiar with the basic definitions of A1-homotopy theory, cf. [14]. Short introductions to those aspects relevant for the obstruction-theoretic torsor classification can be found in papers of Asok and Fasel, cf. for example [2, 4]. The notation in the paper generally follows the one from [2]. We generally assume that we are working over base fields of characteristic ≠2. 3.1. Representability theorem The following A1-representability theorem, significantly generalizing an earlier result of Morel in [14], has been proved in [6]. Theorem 3.1 Let kbe an infinite field, and let X=SpecAbe a smooth affine k-scheme. Let Gbe a reductive group such that each absolutely almost simple component of Gis isotropic. Then there is a bijection HNis1(X;G)≅[X,BNisG]A1between the pointed set of isomorphism classes of rationally trivial G-torsors over Xand the pointed set of A1-homotopy classes of maps X→BNisG. We discuss how this will be applied in the present work. Let k be a field of characteristic ≠2. In the abusive language of Remark 2.7, SO(n) is the special orthogonal group associated to the hyperbolic form of rank n. In particular, it is a semisimple absolutely almost simple group over k which is isotropic (except for n=4 where it is an almost-product of two components with these properties). In particular, combining the representability theorem with Proposition 2.10 provides a bijection between the pointed set of rationally hyperbolic quadratic forms over A and [X,BNisSO(n)]A1. A similar statement applies to the groups Spin(n): being associated to the hyperbolic forms, they are semisimple absolutely almost simple (again, with an exception in case n=4) and isotropic, and this produces a bijection between rationally trivial Spin(n)-torsors and [X,BNisSpin(n)]A1. Note also that the above classification result sets up a bijection between isomorphism classes of torsors and unpointed maps to the classifying space BNisG. When we consider the spin groups which are A1-connected, the corresponding classifying spaces will be A1-simply connected, which implies that there is a canonical bijection between pointed maps X+→BNisSpin(n), the latter pointed by its canonical base point, and unpointed maps X→BNisSpin(n). This is not true for the special orthogonal groups, where the sheaf of connected components is Hét1(μ2), the Nisnevich sheafification of the indicated étale cohomology presheaf. The classification of unpointed maps is obtained by taking a quotient of the pointed maps by the action of the fundamental group sheaf. We will discuss this in Section 6 when we deduce statements concerning quadratic forms from the classification results for Spin(n)-torsors. 3.2. Obstruction theory While the study of A1-homotopy classes of maps into classifying spaces may not seem an easier subject than the torsor classification, the other relevant tool actually allowing to prove some meaningful statements is obstruction theory. The basic statements concerning obstruction theory as applied to torsor classification can be found in various sources, such as [14] or [2, 4]. We only give a short list of the relevant statements which are enough for our purposes. Let (Y,y) be a pointed A1-simply connected space. Then there is a sequence of pointed A1-simply connected spaces, the Postnikov sections (τ≤iY,y), with morphisms pi:Y→τ≤iY and morphisms fi:τ≤i+1Y→τ≤iY such that πjA1(τ≤iY)=0 for j>i, the morphism pi induces an isomorphism on A1-homotopy group sheaves in degrees ≤i, the morphism fi is an A1-fibration, and the A1-homotopy fiber of fi is an Eilenberg–Mac Lane space of the form K(πi+1A1(Y),i+1), the induced morphism Y→τ≤iholimiY is an A1-weak equivalence.Moreover, fi is a principal A1-fibration, that is, there is a morphism, unique up to A1-homotopy, ki+1:τ≤iY→K(πi+1A1(Y),i+2)called the i+1th k-invariant and an A1-fiber sequence τ≤i+1Y→τ≤iY⟶ki+1K(πi+1A1(Y),i+2).From these statements, one gets the following consequence: for a smooth k-scheme X and a pointed A1-simply connected space Y, a given pointed map g(i):X+→τ≤iY lifts to a map g(i+1):X+→τ≤i+1Y if and only if the following composite is null-homotopic: X+⟶g(i)τ≤iY→K(πi+1A1(Y),i+2),or equivalently, if the corresponding obstruction class vanishes in the cohomology group HNisi+2(X;πi+1A1(Y)). If this happens, then the possible lifts can be parametrized via the following exact sequence: [X+,Ωτ≤iY]→[X+,K(πi+1A1(Y),i+1)]→[X+,Ωτ≤i+1Y]→[X+,Ωτ≤iY],where we can also explicitly identify [X+,K(πi+1A1(Y),i+1)]A1≅HNisi+1(X;πi+1A1(Y)). We want to state clearly what this means for the classification of spin torsors or quadratic forms over smooth affine schemes. If we have a torsor for G=Spin(n) or G=SO(n), then the map into the respective classifying space associated by the representability Theorem 3.1 is completely described by a sequence of classes in the lifting sets HNisi+1(X;πi+1A1(BNisG)), which are well defined only up to the respective action of [X+,Ωτ≤iBNisG]A1. Only indices 0≤i+1≤n can appear for schemes of dimension n since the Nisnevich cohomological dimension equals the Krull dimension of X. Conversely, to construct a torsor, one needs a sequence of lifting classes as above, such that the associated obstruction classes in the groups HNisi+2(X;πi+1A1(BNisG)) vanish. Put bluntly, A1-obstruction theory translates questions about A1-homotopy classes of maps (from smooth schemes) into computations of certain (finitely many) cohomology classes. As a result, the classification of Spin(n)-torsors over smooth affine schemes of dimension ≤3 requires only knowledge of the first three A1-homotopy sheaves of BNisSpin(n). This information can be recovered from known computations of A1-homotopy sheaves for special linear and symplectic groups via the sporadic isomorphisms. 3.3. Some A1-homotopy sheaves There are a couple of strictly A1-invariant sheaves of abelian groups which appear in the proofs of the classification results below. We would not recall the detailed definitions here, only provide references where to find such details if required. As a matter of notation, strictly A1-invariant sheaves will usually be denoted by bold letters, like A, Ki, following notational conventions from [2]: Throughout the paper, a couple of K-theory sheaves appear, these are the Nisnevich sheafifications of algebraic K-theory presheaves. The superscript will usually indicate which type of K-theory is referred to: KiQ is Quillen’s algebraic K-theory, KiM denotes Milnor K-theory. Morel’s Milnor–Witt K-theory sheaves will be denoted by KiMW, the definition can be found in [14, Section 2]. Sheafifications of higher Grothendieck–Witt groups are denoted by GWji. For further information concerning their definition and how these sheaves appear as A1-homotopy groups of explicit spaces, cf. [17]. As an exception to this rule, KSpn denotes the symplectic K-groups, represented by the classifying space BSp∞ of the infinite symplectic group, cf. [17]. There are a couple of further sheaves that appear in the work of Asok and Fasel. The sheaves Sn, Tn appear as cokernels of morphisms between K-theory sheaves in [3]. Modifications Sn′ and Sn″ appear in [2]. For the application in A1-obstruction theory, Nisnevich cohomology of sheaves as the above needs to be computed. For details on the Gersten-type complexes computing Nisnevich cohomology and the relevant contraction operation for strictly A1-invariant sheaves, cf. [14] or [2]. The most frequently used formulas for such computations are Bloch’s formula Hd(X,Kd)≅CHd(X)which identifies the Chow groups of a smooth scheme X with the Nisnevich cohomology of the K-theory sheaf Kd (where either Milnor or Quillen K-theory could be used). The standard notation for Chow groups with mod 2 coefficients will be Chd(X)≅Hd(X,Kd/2). There is also a variant of Bloch’s formula, which can be seen as a definition of Chow–Witt groups: Hd(X,KdMW)≅CH˜d(X). Several examples of quadratic forms will be over smooth affine quadrics. We denote by Qd the d-dimensional smooth affine split quadric, cf. [1]. Their Nisnevich cohomology can be easily computed H˜i(Q2d,A)≅{A−d(k)i=d0else,H˜i(Q2d−1,A)≅{A−d(k)i=d−10else,where k is the base field and A−d denotes the d-fold contraction. For the amusement of the reader, we can now compare the low-dimensional A1-homotopy of algebraic groups to classical formulas for homotopy of compact connected Lie groups, cf. [12, Section 3.2]. Example 3.2 The sheaf π0A1 is given by the Whitehead groups, cf. [6, Section 4.3]. For semisimple, simply-connected, absolutely almost simple isotropic k-groups over an infinite field k, this sheaf has trivial contraction, cf. [6, Corollary 4.3.6] and, therefore, [Q2,BNisG]=[Q3,BNisG]=0. This recovers the classical statements that π1(G)=π2(G)=0 for simply connected Lie groups. For groups which are not simply-connected (in the sense of Chevalley groups), we have the finite abelian fundamental group Π (in the sense of Chevalley groups), and an injection π0A1(G)↪Hét1(Π). In the special case G=SO(n), this is the spinor norm map π0A1(SO(n))⟶≅Hét1(μ2). Then [Q2,BNisSO(n)]≅(Hét1(μ2))−1≅μ2recovers the classical statement π1(SO(n))≅Z/2Z, and [Q3,BNisSO(n)]≅(Hét1(μ2))−2≅0again recovers the classical vanishing of π2(SO(n)). Example 3.3 For G semisimple, simply-connected, absolutely almost simple split k-group over an infinite field k, we have π1A1(G)≅{K2MWGsymplecticK2Motherwiseby [20]. Over algebraically closed fields, [Q4,BNisG]≅π1A1(G)−2≅{(K2MW)−2≅GW(k)Gsymplectic(K2M)−2≅Zotherwiserecovers, over algebraically closed fields, the classical computation that π3(G)≅Z for any connected, simply connected compact Lie group. Moreover, [Q5,BNisG]≅π1A1(G)−3≅{W(k)Gsymplectic0otherwiserecovers, over algebraically closed fields, the classical computation of π4(G), which is stated in terms of root system conditions in [12]. Some of these statements about π1A1(G)≅π2A1(BNisG) for the special orthogonal and spin groups will be relevant for the later classification results. 4. Recollections on sporadic isomorphisms In this section, we provide some information on the sporadic isomorphisms identifying the low-rank spin groups with other low-rank groups (for which the relevant low-dimensional A1-homotopy sheaves have already been computed). Since we are interested in stabilization results and the classification of stably hyperbolic forms, we want to obtain more precisely that the sequence of stabilization morphisms for the spin groups from Spin(3) to Spin(6) corresponds, under the sporadic isomorphisms, to the sequence SL2⟶ΔSL2×SL2⟶(2α,2β)Sp4⟶ιSL4,where Δ is the diagonal embedding, (2α,2β) is the embedding arising from the long roots for Sp4 and ι is the natural embedding of Sp4 as stabilizer of the standard symplectic form. This identification of the stabilization morphisms can be done by realizing the usual models of the sporadic isomorphisms inside the six-dimensional quadratic form underlying the identification SL4≅Spin(6). With this goal in mind, parts of the development will differ slightly from the common presentation of sporadic isomorphisms which does not pay respect to the stabilization morphisms. Still, most of the following will be well-known and familiar to many, cf. for example Garrett’s notes [9]. We begin by recalling the identification of SL4 with Spin(6). Consider the four-dimensional k-vector space V=k4 with the natural action of SL4. This induces a natural action of SL4(k) on the six-dimensional space V∧2, that is, a representation SL4→SL6. On V∧2, there is a natural symmetric bilinear form ⟨−,−⟩:V∧2×V∧2→k:⟨v1∧w1,v2∧w2⟩=det(v1,w1,v2,w2).The form is non-degenerate and hyperbolic with an orthogonal basis given by (e1∧e2)±(e3∧e4),(e1∧e3)±(e2∧e4),(e1∧e4)±(e2∧e3).The induced action of SL4 on V∧2 will preserve this form, giving a homomorphism SL4→SO(6). It can be checked via the Lie algebra that the kernel is finite, equal to the subgroup {±1}, hence the homomorphism SL4→SO(6) induces the sporadic isomorphism SL4≅Spin(6). This implies the following: Proposition 4.1 The morphism BSL4→BNisSO(6)induced by the sporadic isogeny SL4→SO(6)is given as follows: if Ris a commutative ring and Pis an oriented projective R-module of rank 4, then the associated quadratic form of rank 6 is given by the projective R-module P∧2equipped with the evaluation form P∧2⊗P∧2→P∧4≅R,where the first map is the projection from the tensor product to the exterior product and the second isomorphism is the orientation of P. Next, we consider the sporadic isomorphism Sp4≅Spin(5) and its relation to the description of Spin(6) obtained above. There is a natural embedding of Sp4 into SL4 as subgroup of matrices preserving a symplectic form on V=k4. The composition with the above identification provides a group homomorphism Sp4→SL4→SO(6) arising from the induced action of Sp4 on V∧2. Viewing the symplectic form on V as a linear form ω:V∧2→k gives a decomposition of the quadratic space V∧2 as direct sum of the five-dimensional quadratic space W=kerω with a line equipped with the standard form x↦x2. Now the action of Sp4 on V∧2 will preserve W=kerω, giving us a morphism Sp4→SO(5). Again, it can be checked using the Lie algebra that this induces an isomorphism Sp4≅Spin(5). We have, therefore, proved the following: Proposition 4.2 The morphism BSp4→BNisSO(5)induced by the sporadic isogeny Sp4→SO(5)is given as follows: let Rbe a commutative ring and let Pbe a symplectic module of rank 2, that is, a projective module Pof rank 4 equipped with a symplectic form ω:P∧2→R. The corresponding quadratic form of rank 5is given by the projective module kerωequipped with the evaluation form kerω↪P∧2⊗P∧2→P∧4≅R. Moreover, the decomposition of the six-dimensional quadratic space V as direct sum of W and a line implies that we can in fact identify the stabilization morphism. Proposition 4.3. There is a commutative diagram where the top horizontal is the natural embedding, the bottom horizontal is the stabilization morphism, and the verticals are the sporadic isomorphisms. Now we will deal with the sporadic isomorphism Spin(4)≅SL2×SL2 and its relation with the isomorphisms discussed previously. If we write the four-dimensional space V with the symplectic form ω as a direct sum of 2 two-dimensional symplectic spaces, the sporadic isomorphism SL2≅Sp2 induces natural embeddings SL2×SL2↪Sp4↪SL4. The first embedding is the one given by the long roots in Sp4. The composite is the embedding of a Levi subgroup of the parabolic subgroup of SL4 preserving the first of the two-dimensional subspaces. We first set up the sporadic isomorphism SL2×SL2≅Spin(4) and then show how this identification fits with the stabilization to Spin(5). The following is the split version of the classical identification of SL2×SL2 via its action on the quaternions. Proposition 4.4. Consider the matrix algebra Mat2×2(k)equipped with the action of SL2×SL2given by (A=(a11a12a21a22),B=(b11b12b21b22),M)↦A·M·B−1.On the matrix algebra, there is a non-degenerate symmetric bilinear form, the modified trace form ⟨X,Y⟩=−tr(X·WYtW−1)where W=(0−110).The corresponding quadratic form is 2det; it is hyperbolic and preserved by the action of SL2×SL2. Therefore, the above action of SL2×SL2on the matrix algebra induces an isomorphism SL2×SL2≅Spin(4). The corresponding morphism BSL2×BSL2→BNisSO(4)of classifying spaces maps an SL2×SL2-torsor to the associated bundle for the above representation. Proposition 4.5. Consider the action of SL2×SL2on V∧2via the composition SL2×SL2↪SL4→SO(6).The action is trivial on the subspace ⟨(e1∧e2)±(e3∧e4)⟩. Equipped with the restriction of the determinant form from V∧2, it is a hyperbolic plane. The map Mat2×2(k)→V∧2given by (1000)↦e4∧e1,(0100)↦e1∧e3,(0010)↦e4∧e2,(0001)↦e2∧e3is a morphism of SL2×SL2-representations and of quadratic spaces which induces an isomorphism and isometry onto its image. With the identifications from Propositions4.4and4.2, the morphism SL2×SL2≅Spin(4)↪Spin(5)≅Sp4induced by the stabilization morphism is the long-root embedding. Proof By direct computation. For instance, the action on e1∧e3 of a pair of matrices (A=(a11a12a21a22),B=(a33a34a43a44))∈SL2×SL2(embedded as indicated as block-diagonal matrix in SL4) is given by e1∧e3↦(a11e1+a21e2)∧(a33e3+a43e4)=a11a33e1∧e3−a11a43e1∧e4+a21a33e2e3−a21a43e3∧e4.Similarly, we get e1∧e4↦−a11a34e1∧e3+a11a44e1∧e4−a21a34e2∧e3+a21a44e3∧e4e2∧e3↦a12a33e1∧e3−a12a43e1∧e4+a22a33e2∧e3+a22a43e3∧e4e2∧e4↦−a12a34e1∧e3+a12a44e1∧e4−a22a34e2∧e3+a22a44e3∧e4.Then it can be checked directly that the given map Mat2×2(k)→V∧2 is equivariant for the actions. With these formulas, one can compute the images of the basis vectors, for example e1∧e2±e3∧e4↦(a11e1+a21e2)∧(a12e1+a22e2)±(a33e3+a43e4)∧(a34e3+a44e4)=(a11a22−a12a21)e1∧e2±e3∧e4.This shows the claim about the invariant subspace. Similarly, one computes the images of the orthogonal basis vectors for the determinant form on V∧2, and the remaining claims about the map Mat2×2(k) are all proven via such computations. Finally, the identification of the stabilization morphism follows from this: the model for Spin(4) given by the conjugation action of SL2×SL2 on the matrix algebra is embedded as four-dimensional subspace of the model for Spin(6) in V∧2, such that the orthogonal complement is a hyperbolic plane. Moreover, on the side of classical groups, the relevant homomorphism is the long-root embedding SL2×SL2↪SL4, and this proves the claim.□ Remark 4.6. Of course the classical branching rules tell us that the restriction of the six-dimensional representation of SL4 to SL2×SL2 is the direct sum of the natural four-dimensional representation (corresponding to the identification with Spin(4)) and a two-dimensional trivial representation. But we need to identify exactly the morphisms on the groups to compute the induced maps on homotopy. Finally, we get to the sporadic isomorphism for the smallest group. Proposition 4.7. Consider the diagonal embedding Δ:SL2→SL2×SL2. Then SL2acts on Mat2×2(k)by conjugation. The action is trivial on the subspace spanned by the identity matrix. The action preserves the matrices of trace 0. The restriction of the modified trace form to the subspace of trace 0 matrices coincides with the trace form ⟨X,Y⟩=tr(X·Y), which is preserved by the action of SL2. This induces an isomorphism SL2≅Spin(3). The induced map BSL2→BNisSO(3)on classifying spaces maps an SL2-torsor to the associated vector bundle for this representation. Proposition 4.8. With the identifications of Propositions4.4and4.7, the map SL2≅Spin(3)→Spin(4)≅SL2×SL2induced by stabilization of the quadratic form is the diagonal embedding Δ. A direct computation of the morphism SL4→SO(6) shows that the composition SL3→SL4→SO(6) induces the hyperbolic morphism BSL3→BNisSO(6), cf. for example the proof of [5, Proposition 2.3.1]. Proposition 4.9. With the identification of Proposition4.4, the composition SL2⟶ι2SL2×SL2→SO(4)induces the hyperbolic morphism BSL2→BSO(4). Proof A direct computation of the action of SL2 on the four-dimensional matrix space shows that it is conjugate to the map SL2→SL4, which sends a matrix M to the block matrix whose two blocks are M and (M−1)t. This proves the claim.□ 5. Sporadic results on spin torsors In this section, we discuss the classification of stably trivial spin torsors over smooth affine schemes of dimension ≤3. The results will be based on the discussion of the sporadic isomorphisms in Section 4. We will explain in Section 6 how the results from the present section translate to the classification of quadratic forms. Note that the sporadic isomorphisms imply that all the spin groups up to Spin(6) are special in the sense of Serre. Therefore, the results below will in fact provide a classification of all Spin(n)-torsors for n≤6 on smooth affine schemes of dimension ≤3. Since there is no difference between Nisnevich- and étale-local triviality of torsors, we omit the indices in Bét=BNis. The various invariants relevant for the classification of the spin bundles will sit in degrees 2 and 3; but the only stable invariant for dimension ≤3 will be the second Chern class in CH2(X). The results below exhibit essentially three different types of examples of stably trivial spin torsors on smooth affine schemes of dimension ≤3. One type of example comes from the changes in π2A1 of the classifying spaces of spin groups in low ranks, where for Spin(3) and Spin(5), we have K2MW and consequently the lifting classes live in CH˜2(X), which has some additional quadratic information not present in CH2(X). Moreover, π2A1(BSpin(4))≅K2MW×K2MW, which means that there are quite a lot stably trivial spin torsors of rank 4. The second type of example for low ranks can be traced to π3A1BSL2, which provides various types of stably trivial spin torsors related to stably free modules; these will already be trivial by stabilization to Spin(6). Finally, the last type of examples are Spin(6)-torsors detected by lifting classes in CH3(X), which become trivial by stabilization to Spin(7). We proceed from higher ranks to lower ranks, analyzing every time the classification of all torsors and which torsors become trivial upon passing to higher ranks. 5.1. Remark on the stable range The first statement to make is that the relevant homotopy groups of classifying spaces of spin groups are stable from Spin(7) on, that is, the natural maps BNisSpin(n)→BNisSpin(n+1) induce isomorphisms on πiA1 for i≤3 and n≥7. Part of this stabilization result was already established in [21, Theorem 6.8]. The relevant locality of BNisSpin(n) are established in [6]. The other half of the stabilization results can be proved as in [21] using the A1-fiber sequence Q2n→BNisSpin(2n)→BNisSpin(2n+1) and the identification of Q2n as motivic sphere from [1]. This brings down the stable range to n≥8. Getting it down to n≥7 uses the octonion multiplication, cf. [5, Corollary 3.4.3]. With this stabilization at hand, it is then clear that there are no non-trivial stably trivial spin torsors for Spin(n), n≥7 over smooth affine schemes of dimension ≤3. The low-dimensional A1-homotopy sheaves for the spin groups Spin(n) with n≥7 are given explicitly as follows, cf. [5, Section 3.4]: π1A1BNisSpin(n)=0,π2A1BNisSpin(n)=K2M,π3A1BNisSpin(n)=K3ind.Since HNis3(X,K3ind)=0, cf. [5, Lemma 3.2.1], there is only one interesting lifting class for the torsor classification which is the second Chern class in HNis2(X,K2M)≅CH2(X). In particular, for a smooth affine scheme X over an infinite field of characteristic unequal to 2, of dimension ≤3, the second Chern class induces a bijection c2:HNis1(X,Spin(7))⟶≅CH2(X). 5.2. Stably trivial torsors of rank 6 We start by analyzing stably trivial Spin(6)-torsors. By the sporadic isomorphism, these are classified in the same way as oriented rank 4 vector bundles, and over schemes of dimension 3 the latter are all determined by their Chern classes. Proposition 5.1. Let kbe an infinite perfect field of characteristic unequal to 2, let X=SpecAbe a smooth affine scheme over kof dimension ≤3. We have the following statements for classification of maps X→BSpin(6): There is an isomorphism π2A1BSpin(6)≅K2M. The first non-trivial lifting class lives in HNis2(X,K2M)≅CH2(X). In particular, if Xhas dimension ≤2, the second Chern class provides a bijection c2:[X,BNisSpin(6)]≅CH2(X). There is an isomorphism π3A1BSpin(6)≅K3Q. Using the identification HNis3(X,K3Q)≅CH3(X), there is an exact sequence H1(X,K2M)⟶Ωk3CH3(X)→[X,BNisSpin(6)]⟶c2CH2(X)→0.Here the map Ωk3is the looping of the third Postnikov invariant. Over an algebraically closed field, the sequence splits and the invariants of a Spin(6)-torsor of the form P∧2are given by the Chern classes of the oriented projective rank 4 module P. Proof We use the identification of Proposition 4.1, whence it suffices to analyse the obstruction theory for maps into BSL4. By [21], the first three A1-homotopy sheaves are stable and equal to the corresponding Quillen K-theory sheaves. The realizability of all lifting classes follows since the obstruction classes would live in degrees above the Nisnevich cohomological dimension of X. The statements made are then direct applications of the obstruction-theoretic formalism, cf. Section 3. For smooth affine 3-folds over algebraically closed fields, it is known, cf. [10] or [2] (Theorem 6.11 in v1 on the arXiv, unfortunately removed from the published paper), that the set of oriented vector bundles is actually identified with CH2(X)×CH3(X) via the Chern classes. Over non-algebraically closed fields, there could be some problems with torsion classes annihilated by the order of the Postnikov invariant k3.□ We need to analyse the stabilization from Spin(6) to Spin(n), n≥7. To deal with the third A1-homotopy sheaf, recall the following computation from [5, Section 3.4]. Proposition 5.2. The induced morphism K3Q≅π3A1BSL4→π3A1BNisSpin(6)→π3A1BNisSpin(7)≅K3indis the natural projection. Corollary 5.3. Let kbe an infinite perfect field of characteristic unequal to 2, let X=SpecAbe a smooth affine scheme of dimension ≤3over k. A Spin(6)-torsor is stably trivial if and only if its second Chern class is trivial. In particular, the stably trivial spin torsors of rank 6 are in fact classified by coker(Ωk3:H1(X,K2M)→CH3(X)). Proof The morphism BSpin(6)→BNisSpin(∞) induces an isomorphism on π2A1, by A1-2-connectedness of Q6≅hofib(BNisSpin(6)→BNisSpin(7)). The stable value of the second homotopy group is π2A1(BNisSpin(n))≅K2M for n≥6. In particular, the lifting class in HNis2(X,K2M)≅CH2(X), which is the second Chern class, is a stable invariant. The projection in Proposition 5.2 induces the zero map CH3(X)≅HNis3(X,K3Q)→HNis3(X,K3ind)=0.This implies that the third Chern class of the rank 6 quadratic form is an unstable invariant, and there is no stable invariant of degree 3 for quadratic forms since HNis3(X,K3ind)=0. Combining the above assertions shows that a Spin(6)-torsor is stably trivial if and only if its second Chern class is trivial. The cokernel claim follows from Proposition 5.1.□ This provides many examples of stably trivial spin torsors over affine 3-folds, compare to a similar class of examples in [5, Example 4.2.3]. Example 5.4. Let X¯ be a smooth projective variety of dimension 3 over C such that H0(X¯,ωX¯)≠0, that is, there is a global non-trivial holomorphic 3-form. Let X be a complement of a divisor in X¯. By [15, Theorem 2] and [8, Proposition 2.1 and Corollary 5.3], the Chow group CH3(X) is a divisible torsion-free group of uncountable rank. In particular, there are uncountably many isomorphism classes of Spin(6)-torsors which are stably trivial. 5.3. Stably trivial torsors in rank 5 The next step is now to analyse the classification of torsors of rank 5 and check which of these become trivial by passage to Spin(6) (or by adding a hyperbolic plane). By the sporadic isomorphism, the relevant information is contained in the symplectic group Sp4. Proposition 5.5 Let kbe an infinite perfect field of characteristic unequal to 2, let X=SpecAbe a smooth affine scheme over kof dimension ≤3. We have the following statements for classification of maps X→BSpin(5): There is an isomorphism π2A1BSpin(5)≅K2MW. The first non-trivial lifting class lives in HNis2(X,K2MW)≅CH˜2(X). In particular, if Xhas dimension ≤2, then the first Pontryagin class (as an invariant in the Chow–Witt ring of BSp4) provides a bijection p1:[X,BSpin(5)]≅CH˜2(X). There is an isomorphism π3A1BSpin(5)≅KSp3, and an exact sequence HNis1(X,K2MW)⟶Ωk3HNis3(X,KSp3)→[X,BSpin(5)]⟶p1CH˜2(X)→0.The invariants are the characteristic classes of the symplectic bundle corresponding to the Spin(5)-torsor. Proof We use the identification of Proposition 4.2. Then it suffices to analyse the obstruction theory for maps into BSp4. By [21], the first four homotopy sheaves of Sp4 are stable and equal the respective symplectic K-groups. This implies the claim on homotopy groups. As in Proposition 5.1, all classes are realizable because the relevant obstructions live above the Nisnevich cohomological dimension of X. The remaining claims are explicit formulations of the obstruction-theoretic statements in Section 3.□ Recall from [2, Proposition 4.16] that we have the following presentation: Ch2(X)⟶Sq2Ch3(X)→HNis3(X,KSp3)→0,that is, the Nisnevich cohomology group is the cokernel of the Steenrod square on mod 2 Chow groups. The corresponding invariant can be non-trivial; classically, it is the invariant α used by Atiyah and Rees to describe complex plane bundles over S6. However, for X a smooth affine 3-fold over an algebraically closed field, we have Ch3(X)=0 because of the unique divisibility of top Chow groups (Roitman’s theorem). In particular, we get the following: Corollary 5.6 Let kbe an algebraically closed field and let Xbe a smooth affine scheme of dimension ≤3over k. Then the first Pontryagin class induces a bijection p1:HNis1(X,Spin(5))⟶≅CH˜2(X). Now we discuss the behavior of the lifting classes under the stabilization morphism. Proposition 5.7 The morphism K2MW≅π2A1BSpin(5)→π2A1BSpin(6)≅K2Minduced by the stabilization homomorphism Spin(5)→Spin(6)is the usual projection K2MW→K2Mgiven by reduction modulo η. The morphism KSp3≅π3A1BSpin(5)→π3A1BSpin(6)≅K3Qinduced by the stabilization homomorphism Spin(5)→Spin(6)is the forgetful morphism. Proof This follows directly from Proposition 4.3 and the fact that the forgetful morphism (from symplectic bundles to oriented vector bundles) is compatible with stabilization.□ Lemma 5.8. Let kbe an infinite perfect field of characteristic ≠2and let Xbe a smooth scheme. Then the morphism HNis3(X,KSp3)→HNis3(X,K3Q)induced by the forgetful morphism is the zero map. Proof Note that any morphism KSp3→K3Q will induce a morphism of Gersten complexes, and we can use that to compute the induced morphism on cohomology. A cycle representing a class in HNis3(X,KSp3) will be a finite sum, indexed by codimension three points x of X, of elements in GW03(k(x))≅Z/2Z, cf. [2, Section 4]. On the other hand, the degree 3 cycle group in the Gersten complex for K3Q will be a direct sum of copies of Z indexed by the codimension three points. The induced map GW03(k(x))→Z must necessarily be the zero map. This shows that any morphism KSp3→K3Q will induce the zero map in degree 3 Nisnevich cohomology.□ Corollary 5.9 Let kbe an infinite perfect field of characteristic unequal to 2, let X=SpecAbe a smooth affine scheme of dimension ≤3over k. A spin torsor of rank 5 over Ais stably trivial if and only if its second Chern class in CH2(X)is trivial. In particular, we have two invariants detecting stably trivial spin torsors of rank 5. The first invariant lives in ker(CH˜2(X)→CH2(X)). If the first invariant vanishes, then there is a secondary invariant which lives in coker(Ωk3:HNis1(X,K2MW)→HNis3(X,KSp3)). A direct consequence of this is that for smooth affine surfaces over quadratically closed fields, stably trivial spin torsors of rank 5 are already trivial because the hypotheses imply CH˜2(X)≅CH2(X). Example 5.10. We can first consider examples related to quadratic information in the Chow–Witt group. Consider the quadric Q4≃S2∧Gm∧2. We have HNis2(Q4,K2MW)≅GW(k).The projection map K2MW→K2M induces the dimension function GW(k)→Z and any element in the kernel will give rise to a stably trivial spin torsor of rank 5 over Q4. This provides an algebraic realization and generalization of the SO(3,2)-bundles over S2 coming from π2BSO(3,2) which are killed by stabilization to SO(3,3). Consider the quadric Q5≃S2∧Gm∧3. We have HNis2(Q5,K2MW)≅W(k).The projection map K2MW→K2M induces the zero map, in particular any element of W(k) gives rise to a stably trivial spin torsor of rank 5 over Q5. This provides an algebraic realization and generalization of the SO(5)-bundles over S5 coming from π5BSO(5). Example 5.11 Let k be an algebraically closed field of characteristic ≠2. We can consider the four-dimensional smooth affine k-scheme X with stably free rank 2 vector bundle constructed by Mohan Kumar [13]. By [22, Section 5], the rank 2 vector bundle is detected by a non-trivial class in ker(CH˜2(X)→CH2(X)). By Corollary 5.9, this non-trivial class will correspond to a non-trivial stably trivial spin torsor of rank 5 over X. It can be constructed by taking the stably free rank 2 module, viewed as a (stably non-trivial) symplectic line bundle, add a trivial symplectic line and view the resulting Sp4-torsor as spin lift of a quadratic form of rank 5 via the sporadic isomorphism. Example 5.12 Interesting examples of stably trivial spin torsors realizing the degree 3 invariant in HNis3(X,KSp3) can be found over higher-dimensional schemes (but still of A1-homotopical dimension 3). For instance, over the base field k, we have HNis3(Q6,KSp3)≅Ch3(Q6)≅H3(Q6;K3M/2)≅Z/2Z.If k is algebraically closed, this corresponds to the classical statement π6(BSO(5))≅Z/2Z. Note also that HNis1(Q6,K2MW)=0, so these examples are not in the image of the looped Postnikov invariant map Ωk3. Over the base field k=C, this is the complex version of the complex plane bundle over S6 detected by the Atiyah–Rees α-invariant. 5.4. Stably trivial torsors of rank 3 and 4 We begin by identifying the lifting classes of Spin(3)-torsors, via the sporadic isomorphism Spin(3)≅SL2. Proposition 5.13 Let kbe an infinite perfect field of characteristic unequal to 2, let X=SpecAbe a smooth affine scheme over kof dimension ≤3. We have the following statements: There is an isomorphism π2A1BSpin(3)≅K2MW. Consequently, the second lifting class for a torsor is the Euler class of the corresponding oriented rank 2 vector bundle in HNis2(X,K2MW)≅CH˜2(X). There are short exact sequences of strictly A1-invariant sheaves 0→T4′→π3A1BSpin(3)→KSp3→0,and 0→D5→T4′→S4′→0,where D5is a quotient of I5and the canonical morphism K4M/12→S4′becomes an isomorphism after 3-fold contraction. There is an exact sequence HNis1(X,K2MW)⟶Ωk3HNis3(X,π3A1BSpin(3))→[X,BSpin(3)]⟶eCH˜2(X)→0.The third lifting class in HNis3(X,π3A1BSpin(3))decomposes into contributions from a class in D5-cohomology, cf. [2], a motivic cohomology class in HNis3(X,K4M/12), and a mod 2 class in HNis3(X,KSp3)≅coker(Ch2(X)⟶Sq2Ch3(X)). Proof The results follow from the identification of Spin(3) with SL2 in Proposition 4.7 together with computations of A1-homotopy groups. Point (1) follows from [14, Theorem 5.39], the description in (2) is obtained in [2, Theorem 3.3, Lemma 7.2]. The description of HNis3(X;KSp3) in terms of Steenrod operations is established in [2, Proposition 4.16].□ Remark 5.14 Using Proposition 4.7, the examples of quadratic forms corresponding to the torsors above can be constructed fairly explicitly. If X=SpecA is a smooth affine scheme and P is an oriented projective module of rank 2 over A, then we can consider its bundle of orientation-preserving automorphisms, which is the principal SL2-bundle over X such that the associated vector bundle for the standard representation is the original module P. If we take the associated vector bundle for the SL2-representation given by conjugation on trace 0 matrices in Mat2×2(k), we get the required quadratic form of rank 3. This sets up a bijection between oriented projective modules of rank 2 and rationally trivial quadratic forms of rank 3. Note that the projective modules of rank 2 can all be obtained by means of the Hartshorne–Serre construction from codimension 2 local complete intersections in X. Corollary 5.15 Let kbe an algebraically closed field of characteristic unequal to 2, and let Xbe a smooth affine scheme over kof dimension ≤3. Then there is a bijection HNis1(X,Spin(3))≅CH˜2(X). Proof Over an algebraically closed field, the top Chow group of a smooth affine scheme is uniquely divisible, by Roitman’s theorem. In particular, Ch3(X) is trivial. This implies that the contribution from KSp3-cohomology vanishes. The unique divisibility of the multiplicative group of an algebraically closed field implies that the group HNis3(X,S4′)≅HNis3(X,K4M/12) is also trivial, cf. [2, Proposition 5.4]. Finally, the restriction of the sheaf I5 to a smooth affine scheme of dimension ≤4 over an algebraically closed field is trivial. This implies that the contribution from D5-cohomology vanishes. The third lifting set HNis3(X,π3A1BSpin(3)) is, therefore, trivial. Since all the higher obstructions vanish because they live above the Nisnevich cohomological dimension of X, any lifting class in HNis2(X,π2A1BSpin(3)) can be uniquely extended to a map X→BSpin(3). The representability Theorem 3.1 provides the required bijection between the second lifting set and the isomorphism classes of rank 3 spin bundles.□ Example 5.16 Any class in CH˜2(X) yields a non-trivial spin torsor of rank 3 over X. Particularly interesting in this situation are those in the kernel of the projection CH˜2(X)→CH2(X). There are examples of such classes over three-dimensional smooth affine schemes over fields of the form k¯(T) as well as examples over four-dimensional smooth affine schemes over algebraically closed fields as discussed in Example 5.11. Example 5.17 Interesting examples of stably trivial torsors realizing the degree 3 invariants can be found over higher-dimensional schemes (but still of A1-homotopical resp. Nisnevich cohomological dimension 3). For instance, over the base field k, we have HNis3(Q6,D5) is a quotient of HNis3(Q6,I5)≅I2(k), HNis3(Q6,K4M/12)≅k×/12, and HNis3(Q6,KSp3)≅Ch3(Q6)≅Z/2Z.If k is algebraically closed, the first two of these vanish and the last one corresponds to the classical statement that π6(BSO(3))≅Z/2Z. Note that the Nisnevich cohomology long exact sequences associated to the short exact sequences of strictly A1-invariant sheaves from Proposition 5.13 reduce to short exact sequences, because Q6 has only non-trivial Nisnevich cohomology in degrees 0 and 3. So any non-trivial class in the above sets actually gives a non-trivial lifting class in HNis3(Q6,π3A1BSpin(3)). Similarly, for Q7, we have HNis3(Q7,D5) is a quotient of HNis3(Q7,I5)≅I(k), HNis3(Q7,K4M/12)≅Z/12Z, and HNis3(Q7,KSp3)≅0 as in the proof of [2, Lemma 7.3].The second item on the list corresponds to the classical fact that π7BSO(3)≅Z/12Z. Example 5.18 One more type of interesting examples related to K4M/12 should be mentioned. If k is an algebraically closed field, the construction of Mohan Kumar, cf. [13], produces a four-dimensional smooth affine scheme X over k(T) which has a non-trivial class in HNis3(X,K4M/12), detected on CH4(X)/3. Mohan Kumar produced a stably free module of rank 3 from this. In [22], a variation of Mohan Kumar’s construction was shown to produce a stably free module of rank 2 over X (which stabilizes to Mohan Kumar’s example). In our context, this rank 2 stably free module produces a non-trivial spin torsor of rank 3 over X. Clearing denominators, we find that there are examples of quadratic forms of rank 3 over five-dimensional smooth affine schemes over algebraically closed fields detected in HNis3(X,K4M/12). Now that we have discussed classification and examples of torsors of rank 3, we turn to the rank 4 case. Proposition 5.19 Let kbe an infinite perfect field of characteristic unequal to 2, let X=SpecAbe a smooth affine scheme over k. There is a bijection HNis1(X;Spin(4))≅HNis1(X;Spin(3))×HNis1(X;Spin(3)). On the level of lifting classes, the stabilization morphism Spin(3)→Spin(4)is given by the diagonal embedding. In particular, every spin torsor of rank 3 which becomes trivial as a spin torsor of rank 4 is already trivial. Proof This follows directly from Propositions 4.4 and 4.8.□ Remark 5.20 It is now straightforward to get examples of rank 4 torsors by taking pairs of the previously discussed Examples 5.16, 5.17 and 5.18 of rank 3 torsors. Now we want to discuss the stabilization to Spin(5). Proposition 5.21 The map πnA1BSL2×πnA1BSL2⟶≅πnA1BSpin(4)→πnA1BSpin(5)⟶≅πnA1BSp4induced by the stabilization morphism Spin(4)→Spin(5)and the sporadic isomorphisms is given by the sum of the stabilization morphisms πnA1BSL2→πnA1BSp4in the symplectic series. The map πnA1BSL2⟶≅πnA1BSpin(3)→πnA1BSpin(5)⟶≅πnA1BSp4induced by the stabilization morphism Spin(3)→Spin(5)is twice the morphism induced from stabilization SL2→Sp4. Proof For (1), it is clear that the morphism is the sum of the restrictions to the individual factors. By Proposition 4.5, the morphism SL2×SL2→Sp4 is given by the long-root embedding. In particular, both morphisms are stabilization morphisms; one by adding an identity matrix in the lower right corner, one by adding an identity matrix in the upper left. The first one is the usual stabilization, and it remains to show that the other one is homotopic to the first one. Put differently, we want to show that the two embeddings SL2→Sp4 via the two choices of long-root embeddings are A1-homotopic. But one of the long-root embeddings is converted into the other by an appropriate conjugation with an element of the Weyl group. The standard representatives of elements of the Weyl group have explicit elementary factorizations (by definition), which provides the required chain of naive A1-homotopies connecting the two long-root embeddings. This shows (1). Statement (2) follows from (1) together with the assertion of Proposition 4.8 that stabilization of spin groups corresponds to the composition SL2⟶ΔSL2×SL2→Sp4. The first map induces the diagonal on A1-homotopy sheaves and the second takes the sum by (1).□ We recall the effect of the symplectic stabilization SL2→Sp4 on A1-homotopy sheaves. Proposition 5.22 The morphism K2MW≅π2A1BSL2→π2A1BSp4≅K2MWinduced from symplectic stabilization SL2→Sp4is the identity, when we identify K2MWof fields with second group cohomology of the discrete groups. The morphism π3A1BSL2→π3A1BSp4≅KSp3induced from symplectic stabilization SL2→Sp4is the natural projection in the exact sequence in point (2) of Proposition5.13. Proof The first one follows from the symplectic stabilization results in [21], the second one follows from the computations in [2].□ Proposition 5.23 The morphism K2MW×K2MW≅π2A1BSpin(4)→π2A1BSpin(5)≅K2MWinduced from orthogonal stabilization Spin(4)→Spin(5)is the sum of the identities on the two factors, when we identify K2MWof fields with second group cohomology of the discrete groups. With this identification, K2MW≅π2A1BSpin(3)→π2A1BSpin(5)≅K2MWinduced from orthogonal stabilization Spin(3)→Spin(5)is multiplication by 2. The morphism π3A1BSL2×π3A1BSL2≅π3A1BSpin(4)→π3A1BSpin(5)≅KSp3induced from orthogonal stabilization Spin(3)→Spin(5)is the sum of the natural projection of Proposition5.13on each of the two factors. Similarly, π3A1BSL2≅π3A1BSpin(3)→π3A1BSpin(5)≅KSp3induced from orthogonal stabilization Spin(3)→Spin(5)is twice the natural projection of Proposition5.13on each of the two factors. Proof This is a combination of Propositions 5.21 and 5.22.□ Remark 5.24 The above statements can, via complex realization, be compared with the classical statements on the stabilization of the homotopy of the (compact) special orthogonal groups, cf. [18]. Classically, we have the following diagram: where the map f is given by (1,1)↦2, (1,0)↦1. The first generator is the one given by the image of the generator from SO(3) (that is, it is realized by the conjugation action of the unit quaternions on themselves), the second generator is given by left multiplication of unit quaternions on all quaternions. The above computations reproduce exactly this picture. Actually, the development of the sporadic isomorphisms in Section 4 can be used to reprove the classical statements in a different manner with significantly less homotopical arguments. Over C, the sequence [Q4,BSO(3)]A1→[Q4,BSO(4)]A1→[Q4,BSO(5)]A1reproduces exactly the classical sequence above by noting that HNis2(Q4,K2MW)≅GW(k), cf. also Example 3.3. The classical description of the generators Q3→SL2 and Q3→SL2×SL2 also follows from the statements in Section 4. We now have all the homotopical information to discuss stabilization of torsors of ranks 3 and 4 and provide some examples. Proposition 5.25 Let kbe an infinite perfect field of characteristic ≠2and let Xbe a smooth affine scheme of dimension ≤3over k: A spin torsor of rank 3 over Xis stably trivial if and only if its lifting class in CH˜2(X)has 2-torsion image in CH2(X). A spin torsor of rank 4 over Xclassified by (γ,δ)∈CH˜2(X)×CH˜2(X)is stably trivial if and only if the class γ+δhas trivial image in CH2(X). Proof Follows directly from Corollaries 5.15 and 5.9 as well as Propositions 5.19 and 5.23.□ We, therefore, get the following examples of stably trivial spin torsors of ranks 3 and 4 related to the lifting class in the second Chow–Witt group. Example 5.26 Let k be a field and let X be a scheme of dimension ≤3 with a non-trivial class α∈CH˜2(X). Then (α,−α)∈HNis2(X,π2A1BSpin(4)) gives a non-trivial stably trivial torsor of rank 4 over X. More complicated examples of stably trivial torsors of rank 3 or 4 arising from the kernel of CH˜2(X)→CH2(X) can be manufactured as in Example 5.11. Finally, we can get examples related to the prime 2. Over R, the complement X of the conic U2+V2+W2=0 in P2 is a smooth affine scheme with CH2(X)≅Z/2Z. We can lift the class of the k-rational point along CH˜2(X)↠CH2(X) and consider the torsor of rank 3 associated to this element, viewed as lifting class in HNis2(X,π2A1BSpin(3)). The resulting bundle will be non-trivial, but stably trivial because its image in the lifting set HNis2(X,π2A1BSpin(6))≅CH2(X) will be twice the generator, by the stabilization results above. Example 5.27 Any combination of the examples of quadratic forms related to degree 3 invariants, cf. Examples 5.17 and 5.18, will result in a stably trivial torsor because the degree 3 lifting classes are not stably visible. However, torsors of rank 3 over smooth affine schemes X of homotopical dimension 3 and with trivial characteristic class in CH˜2(X) will already become hyperbolic by adding a single hyperbolic plane. Any lifting class not related to KSp3-cohomology will be invisible anyway by Proposition 5.23. On the other hand, the KSp3-class after stabilization by a hyperbolic plane will be twice the projection of the class of the associated projective rank 2 module to HNis3(X,KSp3). But the latter is 2-torsion. 6. Spin torsors vs. quadratic forms Now that we have studied in detail the classification of stably trivial spin torsors, we investigate the relation between spin torsors and quadratic forms. The main point is that the classification of rationally hyperbolic quadratic forms can be obtained from the classification of spin torsors by dividing out the action of the fundamental group π1A1BNisSO(n). In particular, most of the stably trivial spin torsors discussed in Section 5 provide examples of non-trivial stably trivial quadratic forms. 6.1. Quadratic forms vs. spin torsors At this point, we aim to say something about the group HNis1(X,SO(n)) of rationally trivial SO(n)-torsors over smooth affine schemes X. To this end, we want to discuss the relation between the two classification problems induced by the natural map [X,BNisSpin(n)]A1→[X,BNisSO(n)]A1.The spin cover Spin(n)→SO(n) is a finite étale (even Galois) map of degree 2 (over a field of characteristic ≠2) and, therefore, by [14, Lemma 6.5], it is an A1-covering space. On the level of homotopy groups, this implies the following statements: first, there are induced isomorphisms πiA1Spin(n)≅πiA1SO(n)for i≥2. Furthermore, since the group Spin(n) is generated by unipotent matrices, it is A1-connected and there is an extension 1→π1A1Spin(n)→π1A1SO(n)→μ2→0.Finally, the spin covering induces a bijection of pointed sets π0A1SO(n)≅Hét1(μ2),where the target denotes the Nisnevich sheafification associated to the presheaf X↦Hét1(X,μ2), cf. Example 3.2. Similar statements for the groups PGLn (in particular for SO(3)) have been discussed in [7, Section 3]. All in all, we have an A1-fiber sequence Bétμ2→BNisSpin(n)→BNisSO(n),which is the restriction of a similar fiber sequence for étale classifying spaces along the natural map BNisSO(n)→BétSO(n). As in [7, Section 3], if X is a smooth affine scheme, mapping X to the above A1-fiber sequence yields an exact sequence of groups and pointed sets [X,SO(n)]A1→Hét1(X,μ2)→[X,BNisSpin(n)]A1→[X,BNisSO(n)]A1.We first want know when the last map is a surjection. Using the relative obstruction theory, a map X=SpecA→BNisSO(n) classifying a rationally hyperbolic quadratic form on A has a spin lift if the sequence of obstruction classes in HNisi+1(X,πiA1Bétμ2) vanishes. At each stage, there is a choice of lifts given by HNisi(X,πiA1Bétμ2). Note that the only non-trivial homotopy groups of Bétμ2 are π1A1Bétμ2≅μ2,andπ0A1Bét≅Hét1(μ2).Since HNisi(X,μ2) is trivial for any smooth affine scheme X and i≥1, the only relevant obstruction and lifting groups are those associated to Hét1(μ2). In particular, a spin lift exists if the obstruction in HNis1(X,Hét1(μ2)) vanishes. One could consider this obstruction class as an algebraic version of the second Stiefel–Whitney class w2 of the quadratic form. In general, there is no reason for the group HNis1(X,Hét1(μ2)) to vanish, for instance we have HNis1(Q2,Hét1(μ2))≅μ2. However, the stabilization morphisms SO(n)→SO(n+1) induce isomorphisms on connected components for all n≥3 and, therefore, the obstruction class in HNis1(X,Hét1(μ2)) is a stable invariant. In particular, it vanishes for stably trivial quadratic forms, meaning that stably trivial quadratic forms will always have spin lifts. In the remainder of the section, we will concentrate on rationally hyperbolic quadratic forms admitting a spin lift. At this point, the exactness of the sequence of groups and pointed sets at the point [X,BNisSpin(n)]A1 means that the image of the map [X,BNisSpin(n)]A1→[X,BNisSO(n)]A1 of pointed sets is given by the orbit set [X,BNisSpin(n)]/Hét1(X,μ2).Reformulated, the isomorphism classes of rationally hyperbolic quadratic forms of rank n over X which admit a spin lift are given as equivalence classes of Spin(n)-torsors by the action of the degree 2 covers of X. 6.2. Action of line bundles We now describe the action of Hét1(X,μ2) more precisely to be able to compute its orbits on the isomorphism classes of spin torsors. First, we consider the cases of rank 3 and 4 and describe the action of degree 2 covers based on the sporadic isomorphisms. Using the sporadic isomorphism Spin(3)≅SL2 from Proposition 4.7, we find that the spin covering Spin(3)→SO(3) can be described as the degree 2 covering SL2→PGL2. In [7, Section 3], the identification [X,BGL2]/HNis1(X,Gm)≅[X,BNisPGL2]was described explicitly: the action of HNis1(X,Gm)≅Pic(X) on [X,BGL2] is given by twisting the rank 2 vector bundles by line bundles, and the PGL2-torsors are then orbits of rank 2 vector bundles by twists with line bundles. Restricting this to SL2-torsors, we get a description of the action of Hét1(X,μ2) on Spin(3)-torsors; the Cartesian square implies that the action of Hét1(X,μ2) on [X,BSL2] factors through the quotient Hét1(X,μ2)↠ker(2:Pic(X)→Pic(X))associated to the sequence 1→μ2→Gm⟶2Gm→1. Explicitly, an element of Hét1(X,μ2) acts on [X,BSL2] by twisting with the associated 2-torsion line bundle. Similarly, the action of Hét1(X,μ2) on Spin(4)-torsors is given by using the sporadic isomorphism Spin(4)≅SL2×SL2: an element of Hét1(X,μ2) acts on [X,BSL2×BSL2] by twisting both rank 2 bundles simultaneously with the associated 2-torsion line bundle. A similar argument can be made for the other two sporadic cases. In rank 6, we have a sequence of degree 2 coverings SL4≅Spin(6)→SO(6)→PGL4. As in [7, Section 3], we can identify the action of line bundles on [X,BGL4] as the twisting. As done above in rank 3, this implies that the action of Hét1(X,μ2) on [X,BNisSpin(6)] is given by twists with 2-torsion line bundles. Under the sporadic isomorphisms, the restriction of this action to Sp4≅Spin(5) deals with the remaining case. 6.3. Result and examples concerning quadratic forms After all the preparations, we are now ready to discuss the isomorphism classes of stably trivial quadratic forms, or more generally rationally hyperbolic quadratic forms which admit a spin lift. We know that the latter are given by the orbit set [X,BNisSpin(n)]/Hét1(X,μ2), where the action is given by twists with 2-torsion line bundles. It remains to revisit the general classification results and specific examples from Section 5 to see what these results say about stably trivial quadratic forms. First, there is a generic remark. Having identified that the action of Hét1(X,μ2) factors through an action of Pic(X),2 there is a number of cases, relevant for our examples, in which the action will be trivial. Our main examples of rationally or stably trivial spin torsors in Section 5 lived over smooth affine quadrics Qn or varieties constructed by Mohan Kumar in [13]. For n≥3, we have Ch1(Qn)=0. The examples of Mohan Kumar for odd primes p are open subvarieties of a hypersurface complement Pp+1⧹Z where Z has degree a power of p. In particular, these will also have trivial Ch1. In these cases, the classification of rationally hyperbolic forms admitting a spin lift agrees with the classification of Spin(n)-torsors. Example 6.1 The above remark applies to Examples 5.10, 5.12, 5.17 and 5.18. In all these cases, we get examples of non-trivial stably trivial quadratic forms. We consider the classification of rationally trivial quadratic forms of rank 6. Proposition 6.2 Let kbe an infinite perfect field of characteristic unequal to 2, let X=SpecAbe a smooth affine scheme over kof dimension ≤3. The action of a 2-torsion line bundle ℓ∈Hét1(X,μ2)on the lifting classes for Spin(6)-torsors is induced from the standard action of Pic(X)on the Chern classes of vector bundles: c2↦c2+6ℓ2c3↦c3+4ℓ3.In particular, rationally hyperbolic quadratic forms of rank 6 over Xare given by orbits of oriented rank 4 vector bundles modulo twists by 2-torsion line bundles. Such a form is stably trivial if the image of its second Chern class in CH2(X)/Pic(X)2equals the orbit of the trivial bundle. Proof The statements about the action being given by twist with a line bundle follow from the previous discussion in Section 6.2. An oriented rank 4 vector bundle is a direct sum of an oriented rank 3 bundle and a trivial line, by Serre’s splitting and the dimension assumption. The following formulas above reflect what happens to the Chern classes of such a bundle under twist with a line bundle of class ℓ: c2↦c2+3c1ℓ+6ℓ2c3↦c3+2c2ℓ+3c1ℓ2+4ℓ3.The term 2c2ℓ can be omitted because ℓ is 2-torsion in our case. The contributions related to c1 vanish because we have oriented bundles.□ Example 6.3 In the case of Example 5.4, we have Pic(X)=Pic(X¯)/[X¯⧹X]. Now Pic(X¯) is an extension of Pic0(X¯) and NS(X). The latter is finitely generated and the former is an abelian variety. In particular, the 2-torsion in Pic(X) is finite. Consequently, Example 5.4 provides uncountably many isomorphism classes of stably trivial quadratic forms of rank 6. Remark 6.4 The composition SL3→SL4→SO(6) of the stabilization and the sporadic isogeny is the hyperbolic morphism, cf. [5, Propositions 2.3.1 and 2.3.4]. By dimension reasons, for a scheme X of dimension ≤3, every quadratic bundle on X is the hyperbolic bundle of some rank 3 oriented vector bundle. In particular, the examples considered above are not only stably hyperbolic, they are all hyperbolic. The technique above applies more generally. If the variety X arises as hypersurface complement in Pn, then the 2-torsion in the Picard group will be finite. If we find ourselves in a situation where one of the lifting groups for stably trivial Spin-torsors happens to be infinite, there will automatically be infinitely many isomorphism classes of stably trivial quadratic forms. Remark 6.5 For the other cases, the action of Pic(X)2 on the lifting groups is not so easy to identify. It seems reasonable to expect that the action CH˜2(X) can be described as follows: there is an exact sequence CH2(X)→CH˜2(X)→HNis2(X,I2)→0.Now the action of an element of Pic(X)2 on CH2(X) is given as in Proposition 6.2, and the action on HNis2(X,I2) should be given by the addition with the image of the class in Pic(X)2 under the boundary map Pic(X)2⊂CH1(X)→HNis2(X,I2). At the moment, I cannot make this more precise, but probably the Hartshorne–Serre correspondence for rank 2 vector bundles allows to identify exactly the action of line bundle twists on oriented vector bundles. Anyway, it is not clear at this point if the Example 5.11 is actually an example of a non-trivial quadratic form (or just an example of an interesting spin torsor) since the lifting classes detecting non-triviality of the spin torsor are 2-torsion and the Picard group has a non-trivial 2-torsion element. Similarly, the action of twisting by 2-torsion line bundles on the cohomology groups HNis3(X,π3A1(BNisSpin(n))) needs to be made explicit to get more detailed results on the classification of rationally hyperbolic quadratic forms. In any case, the combination of the results in Section 5 and the computation in Proposition 6.2 imply the following general result: Proposition 6.6 Let kbe an infinite perfect field of characteristic ≠2, and let X=SpecAbe a smooth affine scheme over kof dimension ≤3. A rationally hyperbolic quadratic form over Ais stably trivial if and only if the image of its second Chern class in CH2(X)/Pic(X)2is in the orbit of the hyperbolic form, where the action of line bundle class ℓis given by x↦x+6ℓ2. It seems that the action of Pic(X)2 would always be an additive action by 2-torsion elements on the Nisnevich cohomology groups. If true, this would generally imply that spin torsors corresponding to lifting classes which are not 2-torsion always give non-trivial rationally hyperbolic quadratic forms. There is one more generic class of stably trivial quadratic forms in rank 4. Proposition 6.7 Let kbe an infinite perfect field of characteristic ≠2, and let X=SpecAbe a smooth affine scheme over kof dimension ≤3. Let α∈CH˜2(X)be a class. Then the lifting class (α,−α)∈HNis2(X,π2A1BSpin(4))induces a non-trivial stably trivial form if αis non-trivial in CH2(X)/Pic(X)2. The quadratic bundles in the proposition are stably hyperbolic, because every bundle of rank 6 is hyperbolic as discussed in Remark 6.4. Using the identification of the hyperbolic morphism in rank 4, cf. Proposition 4.9, we can actually say something about when these bundles are hyperbolic. Proposition 6.8 Let kbe an infinite perfect field of characteristic ≠2, and let X=SpecAbe a smooth affine scheme over kof dimension ≤3. Under the bijection of Proposition5.19and the explicit identification of the hyperbolic morphism in Proposition4.9, a rank 4 spin torsor in HNis1(X,Spin(4))≅HNis1(X,SL2)×HNis1(X,SL2)is hyperbolic if and only if the first component in the product decomposition is trivial. This implies the existence of many stably hyperbolic, but non-hyperbolic quadratic forms of rank 4 over schemes of dimension ≤3. Example 6.9 We can take one of the SL2-torsors discussed in Example 5.17 or 5.18. Let α∈H1(X,SL2) be one such torsor, and let β∈H1(X,SL2) be any other torsor. Then the torsor corresponding to the element (α,β)∈H1(X,SL2)×H1(X,SL2)≅H1(X,Spin(4))will be a non-trivial stably trivial spin torsor, which has no reduction of structure along the hyperbolic morphism SL2→Spin(4). Moreover, by the previous remark, the Picard group for the schemes in Example 5.17 or 5.18 is trivial, implying that the classification of spin torsors and quadratic forms agree in this case. Consequently, the torsor described above corresponds to a stably trivial (hence stably hyperbolic) quadratic form which is not hyperbolic, providing some new examples analogous to [11, Example VIII.2.5.3]. Acknowledgements Originally, the motivation for studying stably trivial quadratic forms arose from my attempt to answer MO-question 166249 ‘Metabolic vs stably metabolic’ by K. J. Moi. The actual work on the project was initiated during a pleasant stay at Wuppertal financed by the DFG SPP 1786 ‘Homotopy theory and algebraic geometry’; and it was finished during a pleasant stay at Institut Mittag-Leffler in the special program ‘Algebro-geometric and homotopical methods’. I would like to thank the anonymous referee for helpful comments which improved the presentation of the results. References 1 A. Asok, B. Doran and J. Fasel, Smooth models of motivic spheres and the clutching construction, Int. Math. Res. Not. IMRN , 2016. 2 A. Asok and J. Fasel, A cohomological classification of vector bundles on smooth affine threefolds, Duke Math. J. 163 ( 2014), 2561– 2601. Google Scholar CrossRef Search ADS 3 A. Asok and J. Fasel, Algebraic vector bundles on spheres, J. Topol. 7 ( 2014), 894– 926. 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Notes, available online at http://www-users.math.umn.edu/~garrett/m/v/sporadic_isogenies.pdf, 2015. 10 N. M. Kumar and M. P. Murthy, Algebraic cycles and vector bundles over affine three-folds, Ann. Math. (2) 116 ( 1982), 579– 591. Google Scholar CrossRef Search ADS 11 M. A. Knus, Quadratic and Hermitian Forms over Rings. Grundlehren der Mathematischen Wissenschaften 294 , Springer, Berlin, 1991. Google Scholar CrossRef Search ADS 12 M. Mimura, Homotopy theory of Lie groups, Handbook of algebraic topology (Ed. I. James), Elsevier, North-Holland, Amsterdam, 1995. 13 N. Mohan Kumar, Stably free modules, Am. J. Math. 107 ( 1985), 1439– 1443. Google Scholar CrossRef Search ADS 14 F. Morel, A1 -Algebraic Topology over a Field, Volume 2052 of Lecture Notes in Mathematics , Springer, Heidelberg, 2012. 15 M. P. Murthy and R. G. Swan, Vector bundles over affine surfaces, Invent. Math. 36 ( 1976), 125– 165. Google Scholar CrossRef Search ADS 16 S. Parimala, Failure of a quadratic analogue of Serre’s conjecture, Am. J. Math. 100 ( 1978), 913– 924. Google Scholar CrossRef Search ADS 17 M. Schlichting and G. Tripathi, Geometric models for higher Grothendieck–Witt groups in A1-homotopy theory, Math. Ann. 362 ( 2015), 1143– 1167. Google Scholar CrossRef Search ADS 18 N. Steenrod, The Topology of Fibre Bundles. Princeton Mathematical Series, 14 , Princeton University Press, Princeton, 1951. Google Scholar CrossRef Search ADS 19 R. G. Swan, K-theory of quadric hypersurfaces, Ann. Math. 122 ( 1985), 113– 153. Google Scholar CrossRef Search ADS 20 M. Wendt, A1-homotopy of Chevalley groups, J. K-Theory 5 ( 2010), 245– 287. Google Scholar CrossRef Search ADS 21 M. Wendt, Rationally trivial torsors in A1-homotopy theory, J. K-Theory 7 ( 2011), 541– 572. Google Scholar CrossRef Search ADS 22 M. Wendt, Variations in A1 on a theme of Mohan Kumar. Preprint, arXiv:1704.00141v1. © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

The Quarterly Journal of Mathematics – Oxford University Press

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